Logarithmics Differentiation Help

Calculate the derivative of the function

F ( x ) = (cos x ) ^{(sin x )} , 0 < x < .

Solution 1

We take the natural logarithm of both sides:

Now we calculate the derivative using the formula (*) preceding this example: The derivative of the left side of (†) is

Using the product rule, we see that the derivative of the far right side of (†) is

We conclude that

Thus

Calculate the derivative of F ( x ) = x ² · (sin x ) · 5 ^x .

Solution 2

We have

Using formula (*), we conclude that

 hence